Interference of one dimensional condensates

1
Interference of one dimensional condensates
Experiments Schmiedmayer et al., Nature Physics
(2005,2006)
Transverse imaging
Longitudial imaging
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Interference between Luttinger liquids
Experiments Hofferberth, Schumm, Schmiedmayer
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Distribution function of interference fringe
contrast
Experiments Hofferberth et al.,
arXiv0710.1575 Theory Imambekov et al. ,
cond-mat/0612011
Quantum fluctuations dominate asymetric Gumbel
distribution (low temp. T or short length L)
Thermal fluctuations dominate broad Poissonian
distribution (high temp. T or long length L)
Intermediate regime double peak structure
Comparison of theory and experiments no free
parameters Higher order correlation functions can
be obtained
4
Calculating distribution function of interference
fringe amplitudes
Method II mapping to inhomogeneous sine-Gordon
model
Imambekov, Gritsev, Demler, cond-mat/0612011
Can be used for 1d systems with arbitrary
boundary conditions and at finite temperature
Can be used to study interference of 2d
condensates
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Inhomogeneous Sine-Gordon models
Limiting cases
Bulk Sine-Gordon model
Boundary Sine-Gordon model
wW
w d(x-x0)
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Inhomogeneous Sine-Gordon models
Expand in powers of g
Coulomb gas representation
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Diagonalize Coulomb gas interaction
Introduce probability distribution function
This is the same probability distribution
function that we need for describing
interference experiments
8
From SG models to fluctuating surfaces
Simulate by Monte-Carlo!
Random surfaces interpretation
This method does not rely on the existence of the
exact solution
9
Distribution function of interference fringe
contrast
Experiments Hofferberth et al.,
arXiv0710.1575 Theory Imambekov et al. ,
cond-mat/0612011
Quantum fluctuations dominate asymetric Gumbel
distribution (low temp. T or short length L)
Thermal fluctuations dominate broad Poissonian
distribution (high temp. T or long length L)
Intermediate regime double peak structure
Comparison of theory and experiments no free
parameters Higher order correlation functions can
be obtained
10
Interference of two dimensional condensates
Experiments Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006)
Probe beam parallel to the plane of the
condensates
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Interference of two dimensional
condensates.Quasi long range order and the BKT
transition
12
z
x
Typical interference patterns
13
Experiments with 2D Bose gas
Hadzibabic et al., Nature 4411118 (2006)
x
integration over x axis
z
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Experiments with 2D Bose gas
Hadzibabic et al., Nature 4411118 (2006)
fit by
Integrated contrast
integration distance Dx
15
Experiments with 2D Bose gas. Proliferation of
thermal vortices Hadzibabic et al., Nature
4411118 (2006)
The onset of proliferation coincides with a
shifting to 0.5!
16
Probing spin systems using distribution function
of magnetization
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Probing spin systems using distribution function
of magnetization
Cherng, Demler, New J. Phys. 97 (2007)
Magnetization in a finite system
Average magnetization
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Distribution Functions
?
?
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Using noise to detect spin liquids
Spin liquids have no broken symmetries No sharp
Bragg peaks
Algebraic spin liquids have long range spin
correlations
A
Noise in magnetization exceeds shot noise
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Summary of part I
Experiments with ultracold atoms provide a new
perspective on the physics of strongly
correlated many-body systems. Quantum noise is a
powerful tool for analyzing many body states of
ultracold atoms
21
Outline
Part I Detection and characterization of many
body states
Part II New challenges in quantum many-body
theory non-equilibrium coherent dynamics
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Landau-Zener tunneling
Landau, Physics of the Soviet Union 346
(1932) Zener, Poc. Royal Soc. A 137692 (1932)
E1
Probability of nonadiabatic transition
w12 Rabi frequency at crossing point td
crossing time
E2
Hysteresis loops of Fe8 molecular clusters
Wernsdorfer et al., cond-mat/9912123
23
Single two-level atom and a single mode field
Jaynes and Cummings, Proc. IEEE 5189 (1963)
Observation of collapse and revival in a one atom
maser
Rempe, Walther, Klein, PRL 58353 (87)
See also solid state realizations by R.
Shoelkopf, S. Girvin
24
Superconductor to Insulator transition in thin
films
d
Superconducting films of different
thickness. Transition can also be tuned with a
magnetic field
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Scaling near the superconductor to insulator
transition
Yes at higher temperatures
No at lower temperatures
Yazdani and Kapitulnik Phys.Rev.Lett. 743037
(1995)
Mason and Kapitulnik Phys. Rev. Lett. 825341
(1999)
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Mechanism of scaling breakdown
New many-body state
Extended crossover
Kapitulnik, Mason, Kivelson, Chakravarty, PRB
63125322 (2001)
Refael, Demler, Oreg, Fisher PRB 7514522 (2007)
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Dynamics of many-body quantum systems
Heavy Ion collisions at RHIC
Signatures of quark-gluon plasma?
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Dynamics of many-body quantum systems
Big Bang and Inflation
Fluctuations of the cosmic microwave background
radiation. Manifestation of quantum
fluctuations during inflation
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GoalUse ultracold atoms to create many-body
systems with interesting collective
propertiesKeep them simple enough to be able
to control and understand them
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Non-equilibrium dynamics ofmany-body systems of
ultracold atoms
1. Dynamical instability of strongly interacting
bosons in optical lattices 2. Adiabaticity
of creating many-body fermionic states in
optical lattices 3. Dynamical instability of the
spiral state of F1 ferromagnetic
condensate 4. Dynamics of coherently split
condensates 5. Many-body decoherence and Ramsey
interferometry 6. Quantum spin dynamics of cold
atoms in an optical lattice
31
Dynamical Instability of strongly interacting
bosons in optical lattices
32
Atoms in optical lattices
Theory Zoller et al. PRL (1998)
Experiment Kasevich et al., Science (2001)
Greiner et al., Nature (2001)
Phillips et al., J. Physics B
(2002)
Esslinger et al., PRL (2004)
Ketterle et al., PRL (2006)
33
Equilibrium superfluid to insulator transition
Theory Fisher et al. PRB (89), Jaksch et al. PRL
(98) Experiment Greiner et al. Nature (01)
Superfluid
Mott insulator
t/U
34
Moving condensate in an optical lattice.
Dynamical instability
Theory Niu et al. PRA (01), Smerzi et al. PRL
(02) Experiment Fallani et al. PRL (04)
Related experiments by Eiermann et al, PRL (03)
35
Question How to connect the dynamical
instability (irreversible, classical) to the
superfluid to Mott transition (equilibrium,
quantum)
Possible experimental sequence
36
Dynamical instability
Wu, Niu, New J. Phys. 5104 (2003)
Classical limit of the Hubbard model.
Discreet GP equation
Current carrying states
Linear stability analysis States with pgtp/2 are
unstable
Amplification of density fluctuations
unstable
unstable
37
Dynamical instability for integer filling
GP regime .
Maximum of the current for .
When we include quantum fluctuations, the
amplitude of the order parameter is suppressed
decreases with increasing phase
gradient
38
Dynamical instability for integer filling
Dynamical instability occurs for
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Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
Phase diagram. Integer filling
Altman et al., PRL 9520402 (2005)
40
Optical lattice and parabolic trap.
Gutzwiller approximation
The first instability develops near the edges,
where N1
U0.01 t J1/4
Gutzwiller ansatz simulations (2D)
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Beyond semiclassical equations. Current decay by
tunneling
Current carrying states are metastable. They can
decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
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Decay rate from a metastable state. Example
Expansion in small e
Our small parameter of expansion proximity to
the classical dynamical instability
44
Weakly interacting systems. Quantum rotor
model. Decay of current by quantum tunneling
At p??/2 we get
For the link on which the QPS takes place
d1. Phase slip on one link response of the
chain. Phases on other links can be treated in a
harmonic approximation
45
For dgt1 we have to include transverse directions.
Need to excite many chains to create a phase slip
Longitudinal stiffness is much smaller than the
transverse.
The transverse size of the phase slip diverges
near a phase slip. We can use continuum
approximation to treat transverse directions
46
Weakly interacting systems. Gross-Pitaevskii
regime. Decay of current by quantum tunneling
Fallani et al., PRL (04)
Quantum phase slips are strongly suppressed in
the GP regime
47
Strongly interacting regime. Vicinity of the
SF-Mott transition
Close to a SF-Mott transition we can use an
effective relativistivc GL theory (Altman,
Auerbach, 2004)
48
Strongly interacting regime. Vicinity of the
SF-Mott transition Decay of current by quantum
tunneling
Action of a quantum phase slip in d1,2,3
Strong broadening of the phase transition in d1
and d2
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Decay of current by quantum tunneling
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Decay of current by thermal activation
phase
j
DE
Escape from metastable state by thermal
activation
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Thermally activated current decay. Weakly
interacting regime
DE
Activation energy in d1,2,3
Thermal fluctuations lead to rapid decay of
currents
Crossover from thermal to quantum tunneling
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Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)
Also experiments by Brian DeMarco et al., arXiv
07083074
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Decay of current by thermal fluctuations
Experiments Brian DeMarco et al., arXiv 07083074
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Adiabaticity of creating many-body fermionic
states in optical lattices
57
Formation of molecules with increasing
interaction strength
Strohmaier et al., arXiv0707.314
Saturation in the number of molecules created is
related to the finite rate of changing interactio
n strength U(t)
58
Formation of molecules with increasing
interaction strength
During adiabatic evolution with increasing
attractive U, all single atoms should be
converted to pairs. Entropy is put into the
kinetic energy of bound pairs.
As U is increased, the excess energy of two
unpaired atoms should be converted to the
kinetic energy of bound pairs.
59
Hubbard model with repulsiondynamics of
breaking up pairs
Energy of spin domain walls
Energy of on-site repulsion
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Hubbard model with repulsiondynamics of
breaking up pairs
Stringent requirements on the rate of change of
the interaction strength to maintain
adiabaticity at the level crossing
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Hubbard model with repulsiondynamics of
breaking up pairs
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Hubbard model with repulsiondynamics of
breaking up pairs
Dynamics of recombination a moving pair pulls
out a spin domain wall
High order perturbation theory
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Hubbard model with repulsiondynamics of
breaking up pairs
N itself is a function of U/t
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Hubbard model with repulsiondynamics of
breaking up pairs
Extra geometrical factor to account for different
configurations of domain walls
Probability of nonadiabatic transition
w12 Rabi frequency at crossing point td
crossing time
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Formation of molecules with increasing
interaction strength
Rey, Sensarma, Demler
Value of U/t for which one finds saturation in
the production of molecules
V0/ER10, 7.5, 5.0, 2.5